
In section  we present the basic of the Chu construction and the local Chu functor.

In section  we present the global Chu functor that corresponds to the Chu construction.

In section  we present the standard and classical boolean Chu representation of \(\mathrm {\mathbf {Top}}\)  and the induced boolean Chu representation of the category of information systems \(\textnormal {\textbf {Inf}}\) .

In section  we present the constructive normal Chu representation of the category of Bishop spaces \(\mathrm {\mathbf {Bis}}\) . This representation of \(\mathrm {\mathbf {Bis}}\)  is the constructive analogue of the aforementioned Chu representation of \(\mathrm {\mathbf {Top}}\) . The notion of a Bishop space is Bishop's constructive, function-theoretic alternative to the classical, set-based notion of a topological space (see [1]}-[2]} and [3]}-[4]}).

In section  and  we give the Chu representation of the category \(P̏(X)\)  of subsets of a set \(X\)  and of the category \(P̏^{\rrbracket \llbracket }(X)\)  of complemented subsets of \(X\) , where \(X\)  is a set equipped with an equality \(=_X\)  and an inequality \(\ne _X\) , respectively. All set-theoretic notions mentioned here are within our reconstruction \(\mathrm {BST}\)  of Bishop's set theory found in [5]} and [6]} (see [7]} and, especially, [8]}).

In section  we introduce the generalised Chu category over a \(\mathrm {ccc}\)  \(C̏\)  and an endofunctor \(\Gamma \)  on \(C̏\) .

In section  we define the generalised global Chu functor that corresponds to the generalised Chu construction.

With the help of the generalised Chu construction we provide a generalised Chu representation of the categories of predicates \(\mathrm {\mathbf {Pred}}\)  and of complemented predicates \(\mathrm {\mathbf {Pred}}^{\ne }\)  in sections  and , respectively.

In section  we introduce the antiparallel Grothendieck construction over a product category and a contravariant \(\mathrm {\mathbf {Set}}\) -valued functor on it, which has the Chu construction as a special case, in case \(C̏\)  is a \(\mathrm {ccc}\) .

